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In this Youtube video, at around 5:10, it shows two graphs, when we have constant increase in voltage on the left, we have constant voltage on the right, but then when voltage stops changing on the left, the voltage drops to zero on the right.

But after that it shows a sinusoidal voltage, at beginning it is rising in the left and its also rising in the right, when at its peak in the left, its like its not changing for a split of second, so looking at the first graph it should be zero for the second graph on the right, but it isn't.

Why do these two graphs follow different logic? It seems we see first derivative on the first graph, and second derivative on second graph

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It should be noted that in a real transformer working in frequencies it is designed for, the second description is correct. I have played with transformers for a few years. I have winded my own one power transformer and 4 audio transformers. From what I have learned and from my experiments, the second description is correct.

There is a fact that may or may not play a role. In a real transformer, both windings are winded on the same side, overlapping each other, sometimes interleaved. They are not winded on different sides of the core separately. If they need to be winded on two sides, both primary and secondary windings are winded on the two sides and overlapping each other. This is to minimize leakage inductance.

The way a transformer work is not that the primary current induces magnetic flux, then magnetic flux induce current in secondary winding. It is more like the secondary winding gets its induced current immediately so the magnetic flux in the core is keeping from changing (much). This is clearly stated in the Lenz's Law ("The current induced in a circuit due to a change or a motion in a magnetic field is so directed as to oppose the change in flux"). If the secondary current always has a phase difference from that in the primary, Lenz's Law can't work.

However, if you keep reducing the frequency you apply to the transformer, at frequencies lower than the lowest frequency the transformer was designed for, the first description may kick in. But this is not what the transformer is designed for and the back EMF may not be enough, so the current in the primary winding may keep increasing and finally overheat and kill the transformer.

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